We
consider
an \(H^2\)  optimal model reduction problem using
the transfer functions \(G\)  of (REF ) and \(G_r\)  of (REF )
to reconstruct a novel reduced model \((A_r, B_r, C_r)\)  of preserving the positivity and interconnection structure better than a given reduced model \((A_r^{(0)}, B_r^{(0)}, C_r^{(0)})\)  in the sense of the \(H^2\)  norm,
where the \(H^2\)  norm of \(G\)  is defined as
\(\Vert G\Vert _{H^2} = \sqrt{\frac{1}{2\pi }\int _{-\infty }^\infty {G(i\omega )}^2_\mathrm {F}\omega }\) .
To this end, we note that input \(u\) , output error \(y-y_r\) , and the difference between \(G\)  and \(G_r\)  have \(\sup _{t\ge 0}\Vert y(t)-y_r(t)\Vert _2 \le \Vert G-G_r\Vert _{H^2} \cdot \Vert u\Vert _{L^2}\) , assuming that
systems (REF ) and (REF ) are both asymptotically stable and input \(u\)  is contained in \(L^2({\mathbb {R}}^m)\) , as explained in [1]}, [2]}.
This inequality indicates that
the maximum output error norm can be expected to become almost zero for any small energy input when
\(\Vert G-G_r\Vert _{H^2}\)  is sufficiently small.
Note that
\(\Vert G-G_r\Vert ^2_{H^2} = 2F(A_r, B_r, C_r) + \Vert G\Vert _{H^2}^2, \) 
